Nonlinear ergodic theorems for amenable semigroups of nonexpansive mappings in Hadamard spaces

We generalize some theorems for continuous representation of an amenable semitopological semigroup S as nonexpansive maps on Hilbert spaces to complete CAT(0) spaces, i.e., Hadamard spaces. For proper Hadamard space, we prove the convergence of asymptotically invariant means of the bounded trajectories. In the nonproper case, adding a suitable condition on the means and assuming S is a locally compact Hausdorff semigroup, we prove Δ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\Delta}$$\end{document} -convergence of asymptotically invariant means of the bounded trajectories.